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arXiv · 2508.07972

Bounded lattice tiles that pack with another lattice

Sigrid Grepstad·Mihail N. Kolountzakis·Emmanuil Spyridakis
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Abstract

Suppose L and M are full-rank lattices in Euclidean space, such that vol(L) < vol(M). Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set F (we can even take F to be a finite union of polytopes) such that F+L is a tiling and F+M is a packing. If we do not require measurability of F it is often possible that a set F can be found tiling with both L and M even when L and M have different volumes, for instance if their intersection is trivial. We also show here that such a set can never be bounded if L and M have different volumes.

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Sigrid Grepstad, Mihail N. Kolountzakis, Emmanuil Spyridakis. 2025-08-11. Bounded lattice tiles that pack with another lattice. https://arxiv.org/abs/2508.07972

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